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------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/master15.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAADzgAAAAYAHgAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwC7wLpAwMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAMAAAAQAkhAPAAAAAAAAAA 7wLpA/////8IAAAA+gIAAAAAAAAAAAAABAAAAC0BAgAHAAAA/AIAAP///wAAAAQAAAAtAQMABAAA ACcB//8cAAAA+wLU/wAAAAAAAJABAAAAAABAAABBcmlhbAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAQAAAAtAQQABAAAAC4BGAAEAAAAAgEBAAUAAAAJAgAAAAIeAAAAMgowAEcBDwAAAENvbXBs ZXggTnVtYmVycwAgABgAJgAZAAoAGAAWAAwAIAAYACYAGQAYAA8AFgAEAAAALgEAABwAAAD7AhAA BwAAAAAAvAIAAAAAAQICIlN5c3RlbQAAAAAAABgAAAAcsRMAAgAAAOQEAAAAAAAABAAAAC0BBQAE AAAA8AEEAAMAAAAAAA== ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0008.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0055.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0056.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0056_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0057.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
We write = i for Ö–1 for simplicity.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0057_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0057_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0058.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
We write = i for Ö–1 for simplicity.
Hence, i2 =3D –1.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0058_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0058_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0059.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
We may write the soluti= on of x2 =3D –r using i.=
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
We write = i for Ö–1 for simplicity.
Hence, i2 =3D –1.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0059_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0059_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0060.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
We may write the soluti= on of x2 =3D –r using i.=
Example A= . Find the solutions of x+ 49 =3D 0
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
We write = i for Ö–1 for simplicity.
Hence, i2 =3D –1.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0060_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0060_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0061.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
We may write the soluti= on of x2 =3D –r using i.=
Example A. Find the solutions of x2 + 49 =3D = 0
Use the square-root method.
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
We write = i for Ö–1 for simplicity.
Hence, i2 =3D –1.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0061_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0061_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0062.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
We may write the soluti= on of x2 =3D –r using i.=
Example A. Find the solutions of x2 + 49 =3D = 0
Use the square-root method.
x2 + 49 =3D 0
x2 =3D –49
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
We write = i for Ö–1 for simplicity.
Hence, i2 =3D –1.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0062_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0062_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0063.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
We may write the soluti= on of x2 =3D –r using i.=
Example A. Find the solutions of x2 + 49 =3D = 0
Use the square-root method.
x2 + 49 =3D 0
x2 =3D –49
x =3D = ±Ö–49
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
We write = i for Ö–1 for simplicity.
Hence, i2 =3D –1.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0063_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0063_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0064.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
We may write the soluti= on of x2 =3D –r using i.=
Example A. Find the solutions of x2 + 49 =3D = 0
Use the square-root method.
x2 + 49 =3D 0
x2 =3D –49
x =3D = ±Ö–49 =3D ±Ö49Ö–1
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
We write = i for Ö–1 for simplicity.
Hence, i2 =3D –1.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0064_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0064_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0065.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
We may write the soluti= on of x2 =3D –r using i.=
Example A. Find the solutions of x2 + 49 =3D = 0
Use the square-root method.
x2 + 49 =3D 0
x2 =3D –49
x =3D = ±Ö–49 =3D ±Ö49Ö–1 =3D ±7i
The equation x<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 =3D –1 does no have any real number solution since the= square of any real number is always non-negative. We make u= p a new imaginary number
        = ;          = ;            &n= bsp;        Ö–1
to be a s= olution of this equation.
We write = i for Ö–1 for simplicity.
Hence, i2 =3D –1.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0065_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAPgGKQIEjI+pUQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0065_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0024.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers,
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0066.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0067.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number.
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0068.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number.
Example B.&nb= sp; 5 – 3i,  6i,  –<= /span>17 are complex numbers
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0069.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number.
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers
for 6i =3D 0 + 6i,
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0070.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number.
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers
for 6i =3D 0 + 6i, –17 =3D –17 + 0i.
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0071.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A = complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number.
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers
for 6i =3D 0 + 6i, –17 =3D –17 + 0i.= 3;
In particular, every real number a is also a complex
because a =3D a + 0i.
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0072.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers.
for 6i =3D 0 + 6i, –17 =3D –17 + 0i.= 3;
In particular, every real number a is also a complex
because a =3D a + 0i.
Complex Numbers<= /span>
(Addit= ion and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex numbers. =
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0073.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers.
for 6i =3D 0 + 6i, –17 =3D –17 + 0i.= 3;
In particular, every real number a is also a complex
because a =3D a + 0i.
Complex Numbers<= /span>
(Addit= ion and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex numbers. =
Example C.
(7 + 4i)= + (5 – 3i)
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0074.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers.
for 6i =3D 0 + 6i, –17 =3D –17 + 0i.= 3;
In particular, every real number a is also a complex
because a =3D a + 0i.
Complex Numbers<= /span>
(Addit= ion and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex numbers. =
Example C.
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0075.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers.
for 6i =3D 0 + 6i, –17 =3D –17 + 0i.= 3;
In particular, every real number a is also a complex
because a =3D a + 0i.
Complex Numbers<= /span>
(Addit= ion and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex numbers. =
Example C.
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0076.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers.
for 6i =3D 0 + 6i, –17 =3D –17 + 0i.= 3;
In particular, every real number a is also a complex
because a =3D a + 0i.
Complex Numbers<= /span>
(Addit= ion and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex numbers. =
Example C.
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i) – (5 – 3i)
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0077.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers.
for 6i =3D 0 + 6i, –17 =3D –17 + 0i.= 3;
In particular, every real number a is also a complex
because a =3D a + 0i.
Complex Numbers<= /span>
(Addit= ion and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex numbers. =
Example C.
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i) – (5 – 3i) =3D 7 + 4i – 5 + 3i
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0078.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form =
        = ;            &n= bsp;    a±bi
where a and b are real numbers, areal part and ±bi is called the imaginary part of the complex number
Example B.  5 – 3i,  6i,  –<= /span>17 are complex numbers.
for 6i =3D 0 + 6i, –17 =3D –17 + 0i.= 3;
In particular, every real number a is also a complex
because a =3D a + 0i.
Complex Numbers<= /span>
(Addit= ion and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex numbers. =
Example C.
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i) – (5 – 3i) =3D 7 + 4i – 5 + 3i =3D 2 + 7i
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0040.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multiplication of complex numbers)
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0079.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0080.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0081.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2       
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0082.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0083.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0083_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqAF0AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACl AXIAgQAAAAAAADMzmQECAwL/hI+py+0Po5y02ouz3igEDoaJR4rmiabqyrYuRZbvEqexN+f6zvf+ UVvJcL2bUcZB/pbMpvN50xgfRMyx2lAqsBsuw/ukBsPk8m74yRyzaXXbAB69t/NunX43d9bWeb6v FyiiZQIWZ5f0d3EIECW458X4pSRZUflYpgjEdSlhqLkICtHpqeiICRdH2si5KoYKy0oIKScaoWo7 MWtZaOratMuqWzeGGPuIdNj6OzmZO/rMNugb7cTYuSwMwnycw9kMyS3nEAxTTXOOrllVHHht+xaU ThcqDc3Xrd0BLqsvJSoOSIiAe17JIjjj3S18jQaew7WwRrt8+2hsSoOw3KZE//NSTVsnsaO3P6Qm IvRnTt0/iuicYRF58B3Mfm56PZv5wte2mcwgrmQ5jp9HY8pObqyHAhvOFt+GJnqKNGhNoCMsLUVZ 8EQ6o1tKwXqJcSnPh3cIcmXi6qpGrWSFLIwFdu0wqFaFWji7JK3YtG62mppqEBW7iT8Lp6TX52qR KbrImYvGTW5Le3fhHfOjmKaaqFKtZj5zReO1Um09G0bsSXNgTHjxSJ58GLVTcp95EBJNEl7puqc7 38vVWkXwVK8pz7UY8y1Vr2xyM619yzGv419rlzC7mzjjiMWX47GbFLoxIeK1Cm8j7tSw0B23e4dh 3EZ5wMIvfwb7Tz1njvOrP/8XNF9/DslXkRQXGRjee9u4MBxd59mX4FH0dSdhLwp+BWB5DbJFYIX7 RRbeVheO2AWDAjo4YIH0zcbchxSySGKMyrHwom2Z4TeehzPylpyMPpJ4YolJ+bZfQ3clpt+PSi4Z RmtN5YhVdJsxSWWVTd4YD06DWWFll15SdF+WDhH2ZZlmehfclhaeyWaby6WJXpBuzklnJmEKVGee ejI5HBEb7glooKCR96eghh7a4XlyIspoozzW52ikkk5KaaWWXopppppuymmnnn4Kaqiijkpqqab+ UOOpqo6a5KquvtpPobDOuud1i9KKq5lq6rDKrbn+mpWOUslaC3XAHjuSmLT/CfucUr4iiyFcugUL XqqPRQketFTt9eyj8YFUS3qSyANct9quOdCV7O2I7VEMeesatVSce2E8SRDKXr5kwtbuReSGYpJJ 8EEgQMEC0KvHrlPapO+NRH7RVWVhaTdLSYoYjDHCZth7r3k+KCxZT/muRjICBkd6cpsKAzJiw40B 3Kp2MMo8c8pU2ewCxgefqWxv74UWW9C4vOtyAjizlPEOBfPc84ot+8WyasQVSzPVJi+9HNY+aO2l Hwv+OA+IU8vm7i9cI70zCGlXcDaVOPqsYGRhH8RvgWavPQPeHLRt9NEL8E2w3nwGJTfYUCPZXDgT sy14Bn733bgFgB+w9OSU/0f+gOUy4iiykhQWCnQ4eDKg+QQHJ0364xeUTnngGLD+9LqVLfkLsf2G rgDmjO8MeOW6SwC7Ab8Hn/vvMeIOr5u2e5YH8Zmn3fvpxrsO/PDT/3398TBZu7m5jmf//OXYA6A6 BcSX7vzVTNvhPRnc/wC+A1y37Xv6xZtuvP3Cx19vqMuboL/x7Q97Otub+fLHvwGW6X8aU18ItEa/ jAVQgdWLQAAnuK32aQuDDuRb0jB4PvQlkHwjfBMDG0g+EZzscfNbW/n+tjrdXbCEGURhD2h4NdSF j4Sw01n8LHdBG7oKh5d7IeSMeMQCWlCG3xPiqoi4P/Dp8IBFXGLmmujEU/9BEYmQ08DZgKg5/XEw i47aovO4KD8XNm6KApQcFMmIKDNmj427o6MbYwhHLRIxhCRU2xg38Mc8HmqMfAxkEwwpSEER8npY QyT83phIQy0SfxS8mSMjGSgO9hB6kLzhJTEJKBAykpOW7CQoJUnD4EHQlC9A4yk59ck2sjJns3zl IHPgQlvqkpatJOUuf6mCWIqvksAsph9r2TrhGTMm79OlK18nPWCOjFet8lMzGfVMNwoTVwCJGcwC doUnCnObuaLGu5g3tH3pEZnLvF3d1hMMWyGLnMWsBvewscF2PuhaQoKGPv9pJGMhyJ8AXaY6rTZQ gha0ngelGJQSt1CD9kpZJBbTYERBZTH2reeixizXTgTKUVtWQgthm1ZIfzmuSNxkbGKw6EkvhR7a 5GGiuzjnSxO5OJVkIXmiu+krcWAWnwq1pUEdqlHfObujKpVDS23qR50KVadFsgAAOw== ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0084.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
The most important com= plex number multiplication formula is the product of a pair = of conjugates numbers.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0085.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
The most important com= plex number multiplication formula is the product of a pair = of conjugates numbers. The conjugate of (a + bi) is (a –= bi) and vice versa.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0086.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
(Conjugate Multiplication)
The most important com= plex number multiplication formula is the product of a pair = of conjugates numbers. The conjugate of (a + bi) is (a –= bi) and vice versa.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0087.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
(Conju= gate Multiplication)
(a + bi)(a – bi) =3D a2 + b2 which is always a <= /span>positive number.
The most important com= plex number multiplication formula is the product of a pair = of conjugates numbers. The conjugate of (a + bi) is (a –= bi) and vice versa.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0088.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
(Conju= gate Multiplication)
(a + bi)(a – bi) =3D a2 + b2 which is always a <= /span>positive number.
The most important com= plex number multiplication formula is the product of a pair = of conjugates numbers. The conjugate of (a + bi) is (a –= bi) and vice versa.
Example E.
 
(4 – 3i)(4 + 3i)
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0089.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
(Conju= gate Multiplication)
(a + bi)(a – bi) =3D a2 + b2 which is always a <= /span>positive number.
The most important com= plex number multiplication formula is the product of a pair = of conjugates numbers. The conjugate of (a + bi) is (a –= bi) and vice versa.
Example E.
 
(4 – 3i)(4 + 3i) =3D 42 = + 32 =3D 25
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0090.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
(Conju= gate Multiplication)
(a + bi)(a – bi) =3D a2 + b2 which is always a <= /span>positive number.
The most important com= plex number multiplication formula is the product of a pair = of conjugates numbers. The conjugate of (a + bi) is (a –= bi) and vice versa.
Example E.
 
(4 – 3i)(4 + 3i) =3D 42 = + 32 =3D 25
(Ö5 – 7i)(Ö5 + 7i)
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0092.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
(Conju= gate Multiplication)
(a + bi)(a – bi) =3D a2 + b2 which is always a <= /span>positive number.
The most important com= plex number multiplication formula is the product of a pair = of conjugates numbers. The conjugate of (a + bi) is (a –= bi) and vice versa.
Example E.
 
(4 – 3i)(4 + 3i) =3D 42 = + 32 =3D 25
(Ö5 – 7i)(Ö5 + 7i) =3D (Ö5)2 + 72 
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0093.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(Multi= plication of complex numbers)
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simpl= ify the result.
Complex Numbers<= /span>
Example D.  (4 – 3i)(2 + 7i)        = ;   FOIL 
        = ;       =3D 8 – 6i + 28i – 21i2        set i2 =3D (-1)
        = ;       =3D 8 – 6i + 28i + 21 =
        = ;       =3D 29 + 22i
(Conju= gate Multiplication)
(a + bi)(a – bi) =3D a2 + b2 which is always a <= /span>positive number.
The most important com= plex number multiplication formula is the product of a pair = of conjugates numbers. The conjugate of (a + bi) is (a –= bi) and vice versa.
Example E.
 
(4 – 3i)(4 + 3i) =3D 42 = + 32 =3D 25
(Ö5 – 7i)(Ö5 + 7i) =3D (Ö5)2 + 72 =3D 54 
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Complex Numbers<= /span>
(Division of Complex Numbers)
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0094.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
(Divis= ion of Complex Numbers)
To divide= complex numbers, we put the division as a fraction,
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0095.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
(Divis= ion of Complex Numbers)
To divide= complex numbers, we put the division as a fraction, then mult= iply the top and the bottom of the fraction by the conjugate= of the denominator.  <= /div>
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0096.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
(Divis= ion of Complex Numbers)
To divide= complex numbers, we put the division as a fraction, then mult= iply the top and the bottom of the fraction by the conjugate= of the denominator.  <= /div>
3 – 2i 
4 + 3i 
Example F.  Simplify
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0096_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA2 AAEAgAAAAAAAAAIGjI+py90FADs= ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0097.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
(Divis= ion of Complex Numbers)
To divide= complex numbers, we put the division as a fraction, then mult= iply the top and the bottom of the fraction by the conjugate= of the denominator.  <= /div>
3 – 2i 
4 + 3i 
Example F.  Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0097_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA2 AAEAgAAAAAAAAAIGjI+py90FADs= ------=_NextPart_01C96700.F2125B40 Content-Location: file:///C:/7102A2B3/2ComplexNumbers_files/slide0098.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
(Divis= ion of Complex Numbers)
To divide= complex numbers, we put the division as a fraction, then mult= iply the top and the bottom of the fraction by the conjugate= of the denominator.  <= /div>
3 – 2i 
4 + 3i 
Example F.  Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i) 
(4 + 3i) 
(4 – 3i) 
(4 – 3i) 
*
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Complex Numbers<= /span>
(Divis= ion of Complex Numbers)
To divide= complex numbers, we put the division as a fraction, then mult= iply the top and the bottom of the fraction by the conjugate= of the denominator.  <= /div>
3 – 2i 
4 + 3i 
Example F.  Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i) 
(4 + 3i) 
=3D
(4 – 3i) 
(4 – 3i)